For the function e-x, the linear approximation around x = 2 isa)(3-x)e-2b)1-xc)[3+2√2-(1+√2)x]e-2d)e-2Correct answer is option 'A'. Can you explain this answer?

Question

For the function e-x, the linear approximation around x = 2 isa)(3-x)e-2b)1-xc)[3+2&radic;2-(1+&radic;2)x]e-2d)e-2Correct answer is option 'A'. Can you explain this answer?

Answer

(neglecting higher power of x)

Answer

The linear approximation around x = 2 for the function e^(-x) is given by:

f(x) ≈ f(2) + f'(2)(x - 2)

To find f(2), we substitute x = 2 into the original function:

f(2) = e^(-2)

To find f'(x), we differentiate the function with respect to x:

f'(x) = -e^(-x)

Then, we substitute x = 2 into the derivative:

f'(2) = -e^(-2)

Now we can substitute these values into the linear approximation formula:

f(x) ≈ e^(-2) + (-e^(-2))(x - 2)

Simplifying this expression, we get:

f(x) ≈ e^(-2) - e^(-2)(x - 2)

Therefore, the correct answer is:

c) [3 - 2(x - 2)]e^(-2)

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